On Generalized Strongly p-Convex Functions of Higher Order

for all x, y ∈ I and t ∈ [0, 1]. 'e notion of convexity is crucial to the solution of many real-world problems. Fortunately, many problems encountered in constrained control and estimation are convex. Since the convexity of sets and functions has been the main object of studies of recent years, in many new problems encountered in applied mathematics, the notion of classical convexity is not enough to reach favorite results [1–4]. Recently, several extensions have been considered for classical convexity such that some of these new concepts are based on extension of the domain of convex function or set to a generalized form [5–7]. Some new generalized concepts in this point of view are pseudoconvex function, quasiconvex functions, invex functions, preinvex functions, B-convex functions, strongly convex functions, and generalized strongly convex functions. 'ere are several fundamental books devoted to different aspects of convex analysis optimization [8–13]. Among them, we mention convex analysis by Rickafaller [11], convex analysis and minimization algorithm by Hiriart-Urruty and Lemerechel [8], convex analysis and nonlinear optimization by Browan and Lewin [9], and introducing lectures on convex optimization by Nesterov [10]. Several inequalities, including some famous inequalities are of Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities, are satisfied by convex functions. 'e following inequality is known as Hermite–Hadamard inequality [14–20]. Let f: I ⊂ R⟶ R be a convex function and let c, d ∈ I with c<d, then the following holds:


Introduction
Some geometric properties of convex sets and functions have been studied before 1960 by great mathematicians Hermann, Minkowski, and Werner Fenchel. e classical convexity is defined as follows: a function f: I ⟶ R is said to be convex function if (1) for all x, y ∈ I and t ∈ [0, 1]. e notion of convexity is crucial to the solution of many real-world problems. Fortunately, many problems encountered in constrained control and estimation are convex. Since the convexity of sets and functions has been the main object of studies of recent years, in many new problems encountered in applied mathematics, the notion of classical convexity is not enough to reach favorite results [1][2][3][4]. Recently, several extensions have been considered for classical convexity such that some of these new concepts are based on extension of the domain of convex function or set to a generalized form [5][6][7]. Some new generalized concepts in this point of view are pseudoconvex function, quasiconvex functions, invex functions, preinvex functions, B-convex functions, strongly convex functions, and generalized strongly convex functions. ere are several fundamental books devoted to different aspects of convex analysis optimization [8][9][10][11][12][13]. Among them, we mention convex analysis by Rickafaller [11], convex analysis and minimization algorithm by Hiriart-Urruty and Lemerechel [8], convex analysis and nonlinear optimization by Browan and Lewin [9], and introducing lectures on convex optimization by Nesterov [10].
Let f: I ⊂ R ⟶ R be a convex function and let c, d ∈ I with c < d, then the following holds: e present paper is organized as follows: in the first section, we will give some basic definitions and some basic properties related to our work. Next, we will derive Hermite-Hadamard-type, Fejér-type and Schur-type inequalities for our definition.
A convex function is called strictly convex if the above inequality holds strictly whenever x and y are distinct points and t ∈ (0, 1).
Definition 1 (see [21]). A function f: R n ⟶ R is said to be quasiconvex if for any x, y ∈ R n and t ∈ [0, 1], we have Definition 2. e interval I is said to be p-convex set if for all x, y ∈ I and t ∈ [0, 1].
Definition 3. Let I be convex set. A function f is said to be p-convex function if the following inequality holds: It can be easily seen that, for p � 1, p-convexity reduced to classical convexity of functions defined on I ⊂ (0, ∞).
In [27], Polyak introduced the strongly convex function as follows.
Definition 5 (see [28]). A function f: I ⟶ R is said to be strongly p-convex function, if for all x, y ∈ I and t ∈ [0, 1].
Definition 6 (see [29]). A function f: I ⟶ R is said to be generalized convex function with respect to η: for all x, y ∈ I and t ∈ [0, 1].
Definition 7 (see [26]). A function f: I ⟶ R is said to be generalized p-convex function, if for all x, y ∈ I and t ∈ [0, 1]. e generalized strongly p-convex function [30] is defined as follows.

Definition 8. A function f is said to be a generalized strongly p-convex function, if
In [31], Lin and Fukushima gave the concept of higher order strongly convex functions and also used it many mathematical programs. Mishra and Sharma [32] derived the Hermite-Hadamard-type inequalities of higher order strongly convex function.
Definition 9 (see [31]). A function f on the convex and closed set I is said to be strongly convex function of higher order if and q is any positive real number. If q � 2, then higher order strongly convex function becomes strongly convex functions Now, in the view of above definitions, we are in the position to introduce new generalization of convexity as follows: Definition 10. A function f is said to be generalized strongly convex of higher order if (12) for x, y ∈ I, with μ ≥ 0 and q is any positive real number Some generalizations of strongly p-convex function of higher order are given in [11] for bifunctions. Definition 11. A function f is said to be a generalized strongly p-convex of higher order if for x, y ∈ I, with μ ≥ 0.

Basic Results
is section is to introduce some basic results related to a generalized strongly p-convex function of higher order.
Definition 13 (see [33]). A function η is said to be additive if η (a, b) + η(c, d) � η(a + b, c + d) for all a, b, c, d ∈ R. Proposition 1. Let f, g: I ⟶ R be two generalized strongly p-convex functions of higher order, then the following statements hold: (i) If η is additive, then f + g: I ⟶ R is a generalized strongly p-convex function of higher order. (ii) If η is nonnegatively homogeneous, then for any λ ≥ 0, λf: I ⟶ R is a generalized strongly p-convex function of higher order.
. . . f n be generalized strongly p-convex function of higher order with μ ≥ 0 on I and α i ≥ 0, then f � α 1 f 1 + α 2 f 2 + · · · + α n f n � n i�1 α i f i . is also generalized strongly p-convex functions of higher order.
Proof (i) Take u � [tx p + (1 − t)y p ] 1/p , then by definition of f and g, we obtain where μ * � 2μ and μ ≥ 0. (ii) Let λ ≥ 0, then by definition, we obtain where μ * � λμ and μ ≥ 0. (iii) It directly follows from (i) and (ii). □ Proposition 2. Let f i : R n ⟶ R for i ∈ I be collection of generalized strongly p-convex functions of higher order. en, supremum function is also a generalized strongly p-convex function of higher order.
Proof. Fix x, y ∈ R n and t ∈ (0, 1). For every i ∈ I, we have which implies in turn that is justifies the convexity of supremum function. □ Proposition 3. Let f i : R n ⟶ R be a generalized strongly p-convex function of higher order, then is also a generalized strongly p-convex function.

Main Results
In this section, we will develop the Hermite-Hadamard-, Fejér-, and Schur-type inequalities for generalized strongly p-convex function of higher order.
Theorem 1 (Hermite-Hadamard-type inequality). Let f: I ⟶ R be a generalized strongly p-convex function of higher order with μ ≥ 0 and p > 0 such that η(., .) is bounded above M η , then the function satisfies the following: Journal of Mathematics Proof. We begin the proof by proving the left-hand side of theorem.
Let f: I ⟶ R be a generalized strongly p-convex function of higher order for all x, y ∈ I, we set t � 1/2, and we have Integrating above equation with respect to t over [0, 1], we have Take x � (ta p + (1 − t)b p ) 1/p and y � (tb p + (1 − t) a, p ) 1/p , we obtain We obtain which is the left-hand side of theorem.
To prove right-hand side, we take x � (ta p + (1− t)b p ) 1/p for all t ∈ [0, 1]. By definition, we have Integrating the above inequality with respect to t over [0, 1], we get After solving this, we get Also in the similar way, we obtain From (A) and (B), we have is implies that which is the right-hand side of theorem. is completes the proof.
Theorem 2 (Schur-type inequality). Let f: I ⟶ R be a generalized strongly convex function of higher order with μ ≥ 0 and η(., .) , be a bifunction, then for all x 1 , x 2 , x 3 ∈ I such that x 1 < x 2 < x 3 and x Proof. Let f be a generalized strongly p-convex function of higher order with μ ≥ 0 and x 1 , x 2 , x 3 ∈ I, then